Yes you can use your HDA as the DHCP server. You turn off the router DHCP server and the HDA will provide DHCP leases to all devices, wireless or hard wired.
You can choose to continue to use your router as your network's DHCP server, however you will need to disable that feature on the HDA. There are also tradeoffs that you should be aware of regarding not using the HDA as the DHCP server.
https://www.amahi.org/faq#is-dhcp-server-required
The speed of your devices connecting to the internet and streaming should be unaffected as this only affects who issues the DHCP lease required to connect your devices to the internet. Speed is dependent on your internet provider.
Hope this helps answer your questions.